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ps3sol_6243_2003

# ps3sol_6243_2003 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Problem Set 3 Solutions 1 Problem 3.1 Find out which of the functions V : R 2 R , 2 (a) V ( x 1 , x 2 ) = x 2 + x 2 ; 1 (b) V ( x 1 , x 2 ) = | x 1 | + | x 2 | ; (c) V ( x 1 , x 2 ) = max | x 1 | , | x 2 | ; are valid Lyapunov functions for the systems (1) x ˙ 1 = x 1 + ( x 1 + x 2 ) 3 , x ˙ 2 = x 2 ( x 1 + x 2 ) 3 ; 2 2 2 2 (2) x ˙ 1 = x 2 x 1 ( x 1 + x 2 ), ˙ x 2 = x 1 x 2 ( x 1 + x 2 ); (3) x ˙ 1 = x 2 | x 1 | , x ˙ 2 = x 1 | x 2 | . The answer is: (b) is a Lyapunov function for system (3) - and no other valid pairs System/Lyapunov function in the list. Please note that, when we say that a Lyapunov function V is defined on a set U , then we expect that V ( x ( t )) should non-increase along all system trajectories in U . In the formulation of Problem 3.1, V is said to be defined on the whole phase space R 2 . Therefore, V ( x ( t )) must be non-increasing along all system trajectories, in order for V to be a valid Lyapunov function. To show that (b) is a valid Lyapunov function for (3), note first that system (3) is defined by an ODE with a Lipschitz right side, and hence has the uniqueness of solutions property. Now, every point ( x 1 , x 2 ) R 2 with x 1 = 0 or x 2 = 0 is an equilibrium of (3). Hence V is automatically valid at those points. At every other point in R 2 , V is 1 Version of October 10, 2003

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2 differentiable, with dV/dx = [sgn( x 1 ); sgn( x 2 )] being the derivative. Hence V ( x ) f ( x ) = x 1 x 2 x 1 x 2 = 0 at every such point, which proves that V ( x ( t )) is non-increasing (and non-decreasing either) along all non-equilibrium trajectories. Below we list the “reasons” why no other pair yields a valid Lyapunov function. Of course, there are many other ways to show that. For system (1) at x = (2 , 0), we have x ˙ 1 > 0, x ˙ 2 < 0, hence both | x 1 | and | x 2 | are increasing along system trajectories in a neigborhood of x = (2 , 0). Since all Lyapunov function candidates (a)-(c) increase when both | x 1 | and | x 2 | increase, (a)-(c) are not valid Lyapunov functions for system (1). For system (2) at x = (0 . 5 , 0 . 5), we have x ˙ 1 > 0, x ˙ 2 < 0, hence both | x 1 | and | x 2 | increase along system trajectories in a neigborhood of x = (0 . 5 , 0 . 5). 2 2 For system (3) at x = (2 , 1), we have x ˙ = (2 , 2), hence both x 1 + x 2 and max( x 1 , x 2 ) are increasing along system trajectories in a neigborhood of x = (2 , 1).
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ps3sol_6243_2003 - Massachusetts Institute of Technology...

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